9.2 KB · ⇊ 5 parts · ⤓ Micron 9.2 KB · ⤓ HTML 79.2 KB

top

| Infinite-order triangular tiling | Infinite-order triangular tiling |

|---|---|

| Poincaré disk model of the hyperbolic plane | Poincaré disk model of the hyperbolic plane |

| Type | Hyperbolic regular tiling |

| Vertex configuration | 3 ∞ |

| Schläfli symbol | {3,∞} |

| Wythoff symbol | ∞ / 3 2 |

| Coxeter diagram | |

| Symmetry group | [∞,3], (*∞32) |

| Dual | Order-3 apeirogonal tiling |

| Properties | Vertex-transitive , edge-transitive , face-transitive |

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.

• Symmetry

• Related polyhedra and tiling

• Other infinite-order triangular tilings

• See also

• References

• External links

A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction.

This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.

| * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } | * n 32 symmetry mutation of regular tilings: {3, n } |

|---|---|---|---|---|---|---|---|---|---|---|---|

| Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyper. | Compact hyper. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |

| | | | | | | | | | | | |

| 3.3 | 3 3 | 3 4 | 3 5 | 3 6 | 3 7 | 3 8 | | 3 12i | 3 9i | 3 6i | 3 3i |

| Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family | Paracompact uniform tilings in [∞,3] family |

|---|---|---|---|---|---|---|---|---|---|---|

| Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | [∞,3] + (∞32) | [1 + ,∞,3] (*∞33) | [1 + ,∞,3] (*∞33) | [∞,3 + ] (3*∞) |

| | | | | | | | | | | |

| | | = | = | = | | | | = or | = or | = |

| | | | | | | | | | | |

| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h 2 {∞,3} | s{3,∞} |

| Uniform duals | Uniform duals | Uniform duals | Uniform duals | Uniform duals | Uniform duals | Uniform duals | Uniform duals | Uniform duals | Uniform duals | Uniform duals |

| | | | | | | | | | | |

| | | | | | | | | | | |

| | V3.∞.∞ | V(3.∞) 2 | V6.6.∞ | V3 ∞ | V4.3.4.∞ | V4.6.∞ | V3.3.3.3.∞ | V(3.∞) 3 | | V3.3.3.3.3.∞ |

| Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | Paracompact hyperbolic uniform tilings in [(∞,3,3)] family |

|---|---|---|---|---|---|---|---|---|---|---|---|

| Symmetry: [(∞,3,3)], (*∞33) | Symmetry: [(∞,3,3)], (*∞33) | Symmetry: [(∞,3,3)], (*∞33) | Symmetry: [(∞,3,3)], (*∞33) | Symmetry: [(∞,3,3)], (*∞33) | Symmetry: [(∞,3,3)], (*∞33) | Symmetry: [(∞,3,3)], (*∞33) | [(∞,3,3)] + , (∞33) | | | | |

| | | | | | | | | | | | |

| | | | | | | | | | | | |

| | | | | | | | | | | | |

| (∞,∞,3) | t 0,1 (∞,3,3) | t 1 (∞,3,3) | t 1,2 (∞,3,3) | | t 0,2 (∞,3,3) | t 0,1,2 (∞,3,3) | s(∞,3,3) | | | | |

| Dual tilings | Dual tilings | Dual tilings | Dual tilings | Dual tilings | Dual tilings | Dual tilings | Dual tilings | | | | |

| | | | | | | | | | | | |

| | | | | | | | | | | | |

| | | | | | | | | | | | |

| V(3.∞) 3 | V3.∞.3.∞ | V(3.∞) 3 | V3.6.∞.6 | V(3.3) ∞ | V3.6.∞.6 | V6.6.∞ | V3.3.3.3.3.∞ | | | | |

A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:

Wikimedia Commons has media related to

Infinite-order triangular tiling

.

• Infinite-order tetrahedral honeycomb

• List of regular polytopes

• List of uniform planar tilings

• Tilings of regular polygons

• Triangular tiling

• Uniform tilings in hyperbolic plane

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)

• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.

• reference-mathworld-hyperbolic-tilingciterefweissteinWeisstein, Eric W. "Hyperbolic tiling". MathWorld.

• reference-mathworld-poincar-hyperbolic-diskciterefweissteinWeisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.